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Friday, November 30, 2012

My department will be looking for a new calculus textbook over the next few months. We used to use Stewart, but there was some discontent, and we switched in part due to the high price. We've been using Briggs for about two years, and are very unhappy with it. So we want to switch again.

I asked a month and a half ago for a good discrete math text, and Josh suggested Discrete Math With Ducks. It looks fun, and was inexpensive compared to what we had been using. I'm excited that my request for suggestions panned out. (Thanks, Josh!) I'll have more to say about that next semester when I start teaching from it.

That was a decision I got to make on my own. The calculus textbook will be a group decision. The rest of the department will want a more conventional textbook than what I might want. I'm willing to work with whatever textbook we use, but I'm dreaming now of writing my own. (That will take a few years...)

What we didn't like in Briggs:

The exercises often jumped too quickly to very hard problems

There's nothing on centroids (until multivariable)

Hmm, I know there's more - I'll have to add to that list next week after our department meeting. I'd like to bring suggestions to the meeting, though. Have any of you used a calculus textbook that you love? Do any of you know of a complete textbook (for Calc I, II, and III, ie going through multivariable calculus) that's under $100?

My department doesn't seem interested in open source textbooks, and the two I used this semester weren't impressive enough for me to want to push it. I love the projects in Boelkins, but that only works if you want to teach through projects. The Guichard made some odd choices. I think any open source book will have more of its own personality than the commercial books. That could be fine, but I haven't seen one yet that will cover all the bases for us.

Sunday, November 25, 2012

This is a topic covered in Calculus II. The textbook explanation is inadequate, and I found nothing good online. So I wrote my own explanation. I now understand it better than I ever did before. (Not surprising, huh? If you're a student, this is an important principle of learning. After you think you understand something, try to write an explanation of it and see how much deeper your understanding can get.)

I'd love to improve this, so please let me know where it's unclear.

Imagine a thin sheet of metal cut in an artistic shape. Is there always a spot where you could hold it balanced on your finger? If there is, can we find that spot? We’ll assume the metal has uniform density. This allows us to treat area as equivalent to weight.

The 1-dimensional Case
To think about this, we first imagine a teeter-totter. We know that two people of the same weight must sit the same distance from the fulcrum (balance point) to balance. We also know that a heavier person must move inward if they want to balance with a lighter person. Experiments show that the weights times the distances from the fulcrum must be equal on the two sides for the teeter-totter to balance. (I wonder if there’s a thought experiment we could do that would convince us this must be true, without the actual experimental evidence.)

Example 1: I weigh 170 pounds and sit 5 feet from the center. My son weighs 75 pounds and sits in front of me, 4 feet from the center. Weights times distances = 170*5 + 75*4 = 1150 feet-pounds. We need someone who weighs 230 pounds to sit 5 feet from the center on the other side. We could write this as: w1*d1 + w2*d2 = w3*d3

If we change our perspective to a number line below the teeter-totter, with 0 at the fulcrum, then the values on the left will be negative. We won’t have the same equality – we’ll have

w1*p1 + w2*p2 = -(w3*p3),

where each d (for distance, always positive) was replaced with a p (for position). This becomes

w1*p1 + w2*p2 + w3*p3 = 0,

given that the fulcrum is at 0. But suppose we don’t know where the fulcrum is? Let’s just put our 0 at the left end, and let the former proper place for the 0 - at the fulcrum - be f. Then the equation becomes

w1*(p1 - f) + w2*(p2 - f) + w3*(p3 - f) = 0,

or w1*p1 + w2*p2 + w3*p3 = w1*f + w2*f + w3*f,

or
w1*p1 + w2*p2 + w3*p3 = f(w1 + w2 + w3).

Let W = the sum of all the weights, then we have

f = (w1*p1 + w2*p2 + w3*p3)/W,

which of course extends from 3 weights and positions to n weights and positions.

On to 2 Dimensions
If we use areas instead of weights, we can look for the fulcrum of the x-values (written as an x with a bar over it) and the fulcrum of the y-values (written as a y with a bar over it). For a finite number of small areas, we would get (the same as above)

xbar = (a1*x1 + a2*x2 + ... + an*xn)/A.

If we imagine a shape formed by the area under a function f (where f has positive y values), between x=a and x=b, sliced into infinitely many vertical strips, with the area of each vertical strip given by height times width = f(x).Δ x, then taking the limit as Δ x goes to 0 gives us

For the y value, we need to notice that the vertical center of each vertical strip is at 1/2 *f(x), and we use this instead of x for the position. So we get

or

Many times the area we're interested in will not be touching the x-axis, and so we need area between a top function, f(x), and a bottom function, g(x). The height of each slice will be (f (x)- g(x)), making the area (f(x) - g(x)).Δ x. The vertical center is now given by averaging f and g. We get:

Thursday, November 22, 2012

Did you know that the U.S. Federal Government has a website called We the People, where you can post petitions?

This may be the first math-related petition I've ever 'signed'.

Implement a Policy for Declassifying Discoveries by NSA Mathematicians

The NSA is the largest employer of mathematicians in the United
States. Currently, the discoveries of those mathematicians in their
official areas of research, being deemed potentially critical to
national security, are indiscriminately classified for an indefinite
period, with limited circumstances for declassification.
It is
requested the White House press the NSA for an expiration policy for the
classification status of non-applied discoveries and instituting an
expiration for gag order patents in the interest of furthering American
academia and industry advancement and in the interest of crediting the
discoveries of our nation's talented NSA employees.

Sunday, November 18, 2012

The textbook we use (Briggs), and I think most of the textbooks I've used in the past (including Stewart and Thomas), introduce anti-derivatives before area under a curve. So they show students the notation for indefinite integrals before showing them the notation for definite integrals. I think this is a BIG mistake.

(Why does it use that funny symbol? Why does it have that dx part at the end? Hard to explain without referencing a connection that hasn't been made yet, isn't it?)

And then we start thinking about areas under curves and use a notation that's almost the same.

∫12 f(x) dx (aka definite integral) means find the area which is between f(x) and the x-axis (area below the axis counts as negative), and between x=1 and x=2.

Seeing almost identical notation and names, we're going to assume that these two act the same in some ways. Students are going to expect anti-derivatives, even though it's area we're talking about here. So it's not much surprise when the Fundamental Theorem of Calculus tells us that to find area we can use anti-derivatives.

Wait! That should be a surprise. It's kind of amazing, isn't it? Derivatives give us slope. Why would going backwards in that process give us area?! Seems to me that's a big one we need to meditate on for a while.

This semester I knew I wanted to connect the new ideas with the position, velocity, and acceleration problems, so I introduced anti-derivatives first. And, I showed the indefinite integral symbol. Oops! I shouldn't have. If I had held off, I believe the meaning of the definite integral would have taken hold better in my student's minds.

Until this semester, I've followed the textbook pretty closely, so my way around this problem has been to introduce the 'Area Function' without using this notation. I found this idea/project in a book put out by the MAA. I've revised it a lot over the years, but the original author, Charles Jones (of Grinnell College) still deserves credit for getting me started in this direction. (I wish I could figure out how to thank him personally, but he doesn't seem to be at Grinnell College these days, and google gives me lots of people with that name.)

I've put a pdf of the project here. If you'd like my Word file, just email me (mathanthologyeditor on gmail).

We've started that project, and it's going well enough, but I realized that if I hadn't introduced the indefinite integral, we'd be better off. Next semester I'll get that right.

Tomorrow we wrap up the project, and I clarify the implications of the Fundamental Theorem. Cool stuff!

Tuesday, November 13, 2012

A
monk climbs from the base of a mountain to its top on the one narrow
path up and down, sleeps in a hut at the top, and then descends again to
her monastery the next day. She leaves at about 6am on both days, and
arrives around 6pm on both days. She stops for a break whenever she
feels like it. Will there be a time of day where she’s at the same spot on both days?

Monday, November 5, 2012

A while back I reviewed You Can Count on Monsters, a delightful book showing monsters built from the prime factorizations of each number 2 through 100. (1 is in the book, but is sad, since it can't be made from primes.)

There are now lots of other takes on this idea. Brent, at The Math Less Traveled,
made some gorgeous factor diagrams back in early October. When he
posted them, many of his readers took them as inspiration to do more.
One made a factor tango. Brent was then inspired to improve on his own diagrams. Here's a partial picture of what's he done:

He says he'll be making posters and t-shirts. I think this would make a great poster for a math classroom. Maybe I'll get copies for some of my colleagues.